Error Becomes Science

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error.”

The lost planet

There is a preliminary irony in this story that deserves its place: the planet about to be lost was sought on purpose, and was found by accident. For years astronomers had suspected that in the suspicious void between Mars and Jupiter — the slot left empty by the arithmetical regularity of the planetary distances — a body must be hiding; and a few months earlier, in an observatory of Lower Saxony, a congregation of specialists had constituted itself with a name out of a novel, the “celestial police,” sharing out the zodiac into districts to be patrolled. Piazzi did not yet belong to it: he was compiling a star catalogue, the most methodical and least adventurous labour of astronomy. On the night of the first of January 1801 — the first night of the new century — from the observatory atop the royal palace of Palermo, he noted a little star that did not figure in the catalogues. The evening after it had moved: it was not a star. The most closely watched district of the sky had been stormed by a cataloguer: the chapter’s first lesson, and the reader of Maskelyne already knows it — discovery loves the methodical. In the following weeks Piazzi followed it with growing care and declining health, gathering a score of positions along an arc of sky a few degrees wide; then illness stopped him, the letters to colleagues on the continent travelled with the slowness of the posts in wartime, and by the time astronomical Europe gave chase the object had slipped into the glare of the Sun.

The first planet discovered from the eighteenth century onward — the body we now call the largest of the asteroids — had lasted six weeks: astronomy possessed a fragment of orbit and a looming embarrassment.

The problem was held to be hopeless. To calculate an orbit the methods of the time wanted abundant and well-distributed observations; here there was a handful, crowded onto a tiny arc, and what is more — the point that interests us — discordant among themselves, like all real observations: no orbit could pass exactly through them all. At Brunswick, a twenty-four-year-old mathematician named Carl Friedrich Gauss left his arithmetical researches for a few months and threw himself at the case. He invented, for the occasion, a new method for determining orbits from few observations; and within the method, as a silent gear, a systematic way of combining incompatible measures — of distilling, from numbers that quarrel, the most trustworthy trajectory. In autumn he sent the astronomers his prediction: look here. It was a region of sky appreciably distant from where all the other calculations pointed. Between December 1801 and the following New Year, von Zach and then Olbers re-aimed their telescopes where the young man of Brunswick said: Ceres was there, a breath from the announced position. Gauss became famous in Europe within weeks [Stigler 1986].

The legend stops at the miracle; the book takes an interest in the gear. For the stroke of Ceres was not a triumph of intuition but of the treatment of error: Gauss had won not because his observations were better — they were everyone’s — but because he had squeezed from imperfect observations more truth than anyone else. The message, for those who could read it, was revolutionary: the way one combines measures is worth as much as the measures themselves. It is the idea of this chapter, and to understand how mature — and how scandalous — it was, one must first look in the face the problem that generations of measurers had refused to look at.

The embarrassment of repeated measures

The reader has met the problem since the second chapter, on the roof of an inn at Barcelona: two honest measures of the same thing never coincide. It is the universal experience of anyone who measures in earnest, and for centuries it was lived as an embarrassment to be hidden rather than a datum to be understood. The practical question was: having ten discordant observations, what does one hand over? And the answers of tradition form a small gallery of ways of not facing the question. One could choose the “best” observation — that of the clearest night, the most trusted instrument, the most authoritative observer — treating the others as poor copies: it is the aristocratic answer, which entrusts the truth to the pedigree of the single datum. One could silently discard the rebel observations, which is Méchain’s answer, and we have seen its human price. Or one could take the average: and the average, which to us seems obviousness itself, had to win its respectability one decade at a time. Tycho Brahe, as early as the sixteenth century, systematically averaged his repeated observations, and it was among the reasons for his legendary precision; but even two centuries later there were men of the first rank convinced that to mix good and bad measures could only “multiply the errors” — so wrote, in substance, even Euler, faced with a problem of celestial mechanics in which the combination of the equations seemed to him a gamble. The idea that errors, in aggregating, could compensate one another rather than accumulate — that the heap was more reliable than the single brick — was not at all obvious: it was, rather, contrary to the whole artisanal instinct of precision, which had always staked on the perfect gesture and not on the crowd of imperfect ones [Stigler 1986].

It must be said, in partial defence of the distrustful, that distrust had its practical reasons. An astronomical measure is not born clean: before it can be compared with the others it must be “reduced” — corrected for the refraction of the air, for the aberration of light, for the flexure of the instrument, for temperature — and every reduction is a calculation that can itself go wrong. The observers of the time were factories of arithmetic before they were of observations, and those who worked in them knew that to average badly reduced data means to knead carefully some spoiled ingredients. The instinct to stake on the single, most careful observation, against the crowd of mediocre ones, was not obscurantism: it was the good sense of one who did not yet possess a theory able to say when the crowd is wiser than the champion — and when not.

In the mid-eighteenth century, two practical problems forced the block. The first was the Moon: Tobias Mayer — the lunar cartographer the third chapter saw rewarded by the Board of Longitude — had to extract three unknowns from twenty-seven discordant equations born of his observations of the librations; and instead of choosing the three “best” equations, as practice wanted, he used them all, grouping them with cunning and averaging them in blocks, with a new argument: more observations increase the precision of the result, instead of diluting it. The second was the Earth: the Ragusan Jesuit Roger Boscovich, grappling with the measurement of an arc of meridian between Rome and Rimini, was the first to formulate an explicit rule for passing a law through data that do not respect it — to choose the line that minimizes the sum of the absolute deviations, taken all together, each with its weight. These were the first public methods for treating disagreement: the discrepancy was beginning to leave the territory of shame and enter that of the algorithm. And Mayer’s method soon made its career to the highest floors: Laplace refined it to tame the most famous enigma of the century’s celestial mechanics, the anomalous accelerations of Jupiter and Saturn, showing in 1787 that the two giants exchange a gravitational debt with a breath of nine centuries — a triumph built entirely on the systematic combination of discordant observations generations old. The machine that would find Ceres again had been running in for half a century. But the decisive step was missing: a universal rule, justified and not merely practised — and on that rule would be consumed the most famous quarrel in the history of measure [Stigler 1986].

The most fruitful quarrel in science

In 1805 Adrien-Marie Legendre, one of the great mathematicians of France, published in an appendix to a work on the orbits of comets a few pages destined to change every quantitative science: the method of least squares. The rule is of a simplicity that can be stated in a sentence: among all the candidate laws, choose the one that minimizes the sum of the squares of the deviations between what the law predicts and what has been observed. To square does two services at once: it cancels the sign — to err by excess or by defect weighs the same — and it punishes large deviations more than proportionally, forcing the law not to abandon any observation too blatantly. Legendre’s exposition was limpid, practical, immediately usable, and the Europe of geodesists and astronomers adopted it with the speed with which one adopts the tools that had always been lacking: within twenty years least squares were, as has been written, the Swiss army knife of every science that measures [Stigler 1986]. The method carried with it, among other things, the elegant solution of a problem the reader has already met in other guises: what to do when the measures to be combined are not of equal quality? One weights them — the observation of the fine instrument counts in the reckoning more than that of the rough one, in inverse proportion to the squares of their respective errors — and the weighted average treats the data as the fourth chapter treated the copies of the metre: none excluded, each with its certificate of reliability sewn onto it. Democracy, but a property franchise.

Then, in 1809, Gauss published his great work on the motion of the celestial bodies and, describing the treatment of the observations, called least squares “our principle,” adding that he had used it since 1795 — ten years before Legendre. Legendre, wounded, answered with a letter of icy dignity: priority is established by publication, not by memory. The dispute poisoned the relations between the two for the rest of their lives, and historians have rummaged through it for two centuries: it is almost certain that Gauss told the truth — too many indications converge, including the silent gear of Ceres — and it is certain that Legendre published first, first gave the method a name and a communicable form, and first delivered it to the community. The book will not award the point, because the lesson lies precisely in not being able to award it: private discovery and public discovery are two different things, and science — the reader knows it from the chapter of Harrison and that of Méchain — lives on what is put in common, not on what lies in drawers.

That Gauss, in any case, did not speak as a theorist in an armchair, the rest of his life tells: from 1818 he spent more than a decade directing in person the triangulation of the kingdom of Hanover, the German chapter of that geodetic stitching of Europe we have met elsewhere — years of inns, signals, and registers, tens of thousands of angles measured and adjusted with his least squares. Dissatisfied with the traditional signals, he even invented the instrument symbol of the enterprise, the heliotrope: a little mirror precisely oriented that throws the sun’s reflection to a colleague fifty kilometres away — a lighthouse without a fire, lit by astronomy. The greatest theorist of error was therefore also a field measurer, with his boots in the mud of the Lüneburg; and his theory carries the marks of that experience, for it treats errors not as abstract entities but as colleagues at work: inevitable, troublesome, and — if well organized — extraordinarily productive.

There remained to understand why the rule worked. Legendre had offered it as a practical criterion, elegant and manageable; it was Gauss who gave it its foundation, with a reasoning that overturns the problem: instead of asking which rule is right, let us ask how the errors must behave for the most natural rule — the average — to be right. The answer led him to a curve. (Years later, as a seasoned geodesist, he would give the method a second justification, soberer and still more robust, which asks of the errors no particular form but only that they favour no sign: among all the honest linear recipes, least squares remain the one that wavers least. It is the result the manuals call today, with the customary respect for chronology, the Gauss-Markov theorem.)

The form of error

Here the book reaches its one equation in solemn form, and introduces it as it deserves: in words before in symbols. Imagine measuring a thousand times the same magnitude — the latitude of Barcelona, the mass of a cylinder — and arranging the results along a line, stacking them: the values near the centre will soon form a hill, the distant values will grow ever rarer, and the hill will have two notable properties. It is symmetrical: the errors by excess and by defect, neither having more right than the other, present themselves with equal frequency. And it has steep shoulders and thin tails: small mistakes are ordinary business, large ones are rare, enormous ones rarest of all — but never impossible. The curve that draws this profile is the bell the world calls Gaussian, and which history, with typical Stiglerian justice, ought to call in at least three ways: Abraham de Moivre met it first in 1733, as an approximation of games of chance; Laplace made of it the instrument of his great theorem; Gauss, in 1809, tied it to errors of measure, showing that if the average is the right estimate, then the errors must distribute themselves thus — and that under this curve least squares are not one criterion among many, but the absolutely optimal strategy. Its modern expression, with $\mu$ to indicate the centre of the bell — the value around which the measures crowd — and $\sigma$ its width, that is, the typical size of the error, is:

$$f(x) \;=\; \frac{1}{\sigma\sqrt{2\pi}}\; e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}$$

The reader who does not frequent formulas loses no thread: everything the formula says is already written in the hill — symmetry about the centre, tails that die quickly, and a single number, $\sigma$, to sum up how much the instrument stammers. But whoever looks at it closely finds in it the little scandal that made the method’s fortune: within the law of error appear $\pi$, the number of circles, and $e$, the number of growth — as if chance, rummaged through deeply enough, gave back the noblest constants of mathematics. It is exactly the impression that thunderstruck the nineteenth century, and that Galton’s epigraph delivers without shame: in the irregularity of measures there dwells no chaos; there dwells an order, with a precise and calculable form.

As for the width $\sigma$, the nineteenth century handled it through a cousin of eloquent name: the probable error, the radius within which the coin of error falls heads or tails with equal frequency. From Bessel on, no serious astronomical result came out naked: beside the value, its probable error — the first time in history the margin of ignorance became an obligatory part of the result, with a ”±” symbol serving as conjunction between what is known and how much it is known. The standard deviation, which is the heir of the probable error with a late-nineteenth-century name, says for that matter something marvellously concrete: within one width $\sigma$ of the centre fall about two measures in three; within two widths, nineteen in twenty. The bell is not merely a form: it is a table of expectations, and he who measures consults it as the navigator consults the tides.

The year after, Laplace closed the circle with the result we now call the central limit theorem: every effect born of the sum of many small independent causes — and the error of measure is the textbook case: a breath of air, a friction, a tremor, a refraction — tends to distribute itself in a bell, whatever the nature of the single causes. The Gaussian was therefore not a felicitous hypothesis: it was what the multitude produces by mathematical necessity. And from here descends the moral theorem of the whole chapter, the one that overturns two thousand years of artisanal instinct: precision ceases to be a property of the single gesture and becomes a property of the method. The average of a hundred mediocre measures beats the solitary measure of the virtuoso, and the theory says even by how much: the uncertainty of the average shrinks as $1/\sqrt{n}$ — to quadruple the observations is to halve it. Borda’s repeating circle, which the second chapter saw multiply the readings to temper their mistakes, and the three chronometers of the third, and Maskelyne’s pairs of blind computers: all the practical wisdom accumulated by a century of measurers received, at a stroke, its mathematical patent [Stigler 1986].

The theory put order also in the most scabrous of the rites, the one that had ruined Méchain: the rejection of rebel observations. To throw out a datum because it “came out badly” is the antechamber of fraud; to keep it at all costs may mean letting a gust of wind decide an orbit. The bell offered for the first time a non-confessional criterion: since it says how rare large deviations are, it allows one to ask of every suspect measure, “with what frequency does your size happen by chance?”, and to discard it only when the answer is “hardly ever” — a process with a declared threshold, famous already by mid-nineteenth century in the versions codified by the American astronomers. The rejection criteria remain to this day delicate and debated matter, but the point of civility is acquired: the observation takes its leave with a written justification, it is not made to vanish from the registers. What Méchain buried in a coded notebook, half a century later would be filed with a stamp and a formula: acquitted for improbability.

Calibrating men

The new science of error soon had occasion to exercise itself on the most delicate of its instruments. In 1796 the Astronomer Royal Maskelyne — Harrison’s rival, whom we find here in the role of involuntary founder of a discipline — dismissed his assistant Kinnebrook for a fault then without a name: his transit were systematically late by nearly a second with respect to his principal’s, and no reprimand corrected the drift. For Maskelyne it was negligence; thirty years later, the German astronomer Bessel reopened the file with a new suspicion, and comparing methodically the transit times of different observers — himself included — discovered that all differ from all: each observer carries in his timing a delay of his own, stable, measurable, which is not a fault but a physiology. They called it the “personal equation,” and the observers of the world took to calibrating one another as one calibrates thermometers: the difference between the eye of A and the eye of B, measured and tabulated, was subtracted from the registers like any instrumental correction. Poor Kinnebrook was rehabilitated by statistics a century late; and from the personal equation — from measuring how long a brain takes to register an event — would sprout, at the end of the nineteenth century, experimental psychology: the last science born of a quarrel between astronomers [di) 1995]. Technology, meanwhile, was working to circumvent the problem at the root: the electric chronograph of the mid-nineteenth century — the telegraph lent to the observatory — allowed the transits to be recorded by pressing a key that marks the instant on a tape, replacing the ancient art of counting the clock’s beats by ear with a simple reflex of the finger; the personal equation shrank by an order of magnitude, without ever vanishing entirely. It is the dialectic the reader now recognizes: first the defect is discovered, then it is measured, then the machine is built that narrows it — and meanwhile the defect, from a scandal, has become a chapter of the manual.

The episode deserves its moral, for it is the most radical point of the whole reversal. The first chapter recounted a world in which man was the measure of all things; here the motion is completed the other way round: man becomes one of the things measured — an instrument within the chain, with his drift, his certificate, his equation. The perfect measurer, idol of every artisanal tradition, leaves the stage forever; he is replaced by the characterized measurer, whose defects are known with precision. It is not a degradation: it is a promotion to the dignity of instrument — and the reader who remembers the rate of the chronometers already knows that, in metrology, a constant and known defect is worth as much as perfection.

The curve conquers the world

Once discovered in the stars, the bell revealed itself everywhere. The decisive step — and the most fraught with consequences — was taken in the thirties of the nineteenth century by the Belgian astronomer Adolphe Quetelet, who had the idea of aiming the instruments of celestial mechanics at society: he gathered the chest measurements of thousands of Scottish soldiers, stacked them as one stacks the observations of a star, and saw emerge the same identical symmetrical hill as the astronomical observers. The reading he gave of it made an epoch: if a thousand chests distribute themselves like a thousand measures of a single magnitude, then — he argued — it is as if nature aimed at a model, its “average man,” and the real individuals were the errors of aim. From this vertiginous metaphor was born social statistics: averages, deviations, and regularities applied to births, crimes, suicides, statures — the discovery, intoxicating for the nineteenth century, that society has numerical laws as stable as the heavens. And the bell revealed itself at once also a formidable detector of lies. Examining the statures of the French conscripts, Quetelet noticed that the hill, elsewhere impeccable, presented a suspicious dent: too many young men just below the minimum height for the levy, too few just above — the deformed bell denounced, with the cold elegance of a histogram, how many lads had had themselves measured a little stooped before the recruiting officer. The curve of errors, born to absolve the honest observers, made its debut as an instrument for catching the dishonest declarers: from then on statistics does also the trade of the muhtasib, and inspects the measures of others in search of dented hills — from corporate balance sheets to electoral results. The bell passed therefore from the errors of the astronomers to men of flesh and blood, and changed meaning in the journey: no longer the signature of the imperfection of measures, but the very form of the variability of the world [Stigler 1986].

A caution is fitting, which Quetelet’s critics raised early and which remains current: the translation is not innocent. The error of measure has a true value to pursue; the stature of men does not — biological variability is not a mistake with respect to a model, and “the average man” is an arithmetical fiction useful only until one mistakes it for an ideal. The nineteenth century mistook it often, and the vocabulary itself bears the bruises: to call the distribution “normal,” and thus implicitly abnormal what dwells in the tails, was a lexical choice for which statistics has never finished paying the cultural consequences.

From there the curve never stopped. The artillerists took it, who of the target are the literal professionals: nineteenth-century firing tables ceased to promise trajectories and took to describing patterns — the dispersion of the shots around the aimed point, Gaussian for the same reasons as the observatory errors, a thousand small independent causes among powder, wind, and barrel — and gunnery became an application of the theory of errors, with its probabilities of hitting calculated before firing. Maxwell took it — the reader left him among the coils of the previous chapter — and carried it into matter: the velocities of the molecules of a gas distribute themselves in a bell, and physics discovered that its most solid laws, temperature and pressure foremost, are statistical averages over swarms of random motions — macroscopic order as the visible face of counted disorder. Galton, Darwin’s cousin, pursued it into biological inheritance with the ingenuity of a maker of toys — his “quincunx,” a vertical board in which a rain of pellets bounces off rows of staggered pins and accumulates at the bottom, makes the bell be born before the spectator’s eyes, a mechanical demonstration of Laplace’s theorem that still enchants in science museums — and found in it two new tools. Measuring fathers and sons he discovered that the sons of the tallest fathers are tall, but on average less than the fathers, and the sons of the smallest, small but less so: “regression toward the mean,” which is not a mysterious force but pure arithmetic of the bell, and which has ever since deceived anyone who mistakes for merit or punishment what is only a return toward the centre — the coach who believes the scolding effective after the dreadful performance, the investment fund celebrated after the golden year. And to quantify how much two magnitudes hold hands he invented correlation, the tool with which one studies today anything at all, from drugs to markets. He venerated the curve with the enthusiasm of the epigraph at the head of this chapter, going so far as to write that the Greeks, had they known it, would have deified it. Honesty obliges one to complete the portrait: that same Galton put those tools at the service of the eugenic programme of which he was founder and eponym, and the statistics of human populations was born intertwined with that mortgage, which would cost it — and would cost us — morally very dear. It is the reminder this book must leave to quantitative enthusiasms: to measure men is never a neutral act, and the first chapter had said it in its own way — he who controls the standard controls what is poured into it. On the opposite pan of the moral balance stands, in the same decades, Florence Nightingale, who came back from the Crimean War convinced that to count the dead by cause — and to draw the counts in diagrams a minister could not pretend not to understand — was a religious duty before an administrative one: statistics as a form of organized compassion. The two faces of the curve, the register that discriminates and the register that saves, were already both visible to whoever cared to look.

From error to uncertainty

There remained the last step, the subtlest, and it occupied metrology for much of the twentieth century: to distinguish the two faces of error, and then to dismiss the word itself. The bell governs the random errors — the democratic tremor that repetition tempers according to the law of the root of $n$. But there exists the other error, the one that does not tremble: the crooked instrument, the uncorrected refraction, the plumb line deflected by the mountains — the systematic deviation, which presents itself identical at every measure and which no average will ever dilute, because the average of a thousand equally crooked measures is a crooked conclusion held with great confidence. The distinction lets itself be drawn on a target, and it is the image with which every metrologist carries it in his pocket. A tight but off-centre grouping of shots is the precise and inaccurate marksman: very steady hand, crooked sight — small random error, large systematic; a centred but wide grouping is the accurate imprecise: right aim, trembling hand. Repetition cures the tremor, because the centre of the grouping is estimated ever better; it does not cure the sight, because no average of crooked shots straightens the barrel. They are two independent virtues, paid with different coins — patience the one, self-criticism the other — and to confuse them is the most ancient error of the craft: a long career of concordant measures may be a long career of concordantly wrong ones. The reader already possesses both figures: the rate of the chronometer was a tamed systematic — constant, hence measurable, hence correctable — and the three seconds of arc of Barcelona were an unknown systematic, which Méchain tortured as a fault because his age offered him no conceptual pigeonhole in which to file it. The posthumous rehabilitation of the astronomer, begun by Delambre out of pity and completed by the moderns with calculation, is the ripe fruit of this distinction: his two Barcelonas differed for real and stable causes, not for unworthiness — and the difference, today, would simply be published, with an estimate.

The art of declaring the margin, for that matter, had had a precursor so early that it is worth saluting him: already in 1786 Laplace had estimated the population of France — which no complete census could then count — by multiplying the registered births by a ratio measured on a sample of parishes, and had accompanied the result with the explicit calculation of the probabilities of erring beyond a given threshold: a national number delivered to the government together with its own uncertainty, a century and a half before the polls made the idea familiar. Statistical modernity is not the audacity of the estimate: it is the honesty of the margin that accompanies it.

For it is here that the twentieth century carried to its end the moral revolution of the chapter: the mature measure no longer promises the “true value,” it declares its own margin. Every serious result presents itself today in two numbers — the estimate and the uncertainty, the position of the hill and its width — and the second number is not a confession of weakness but the most worked part of the craft: to say how much one does not know is a knowing, the hardest. Particle physics has made of it even a rite of passage: no discovery is announced below “five sigma,” that is, until the deviation from nothing exceeds five widths of the bell — a threshold so severe that chance crosses it only a few times in ten million — and it is the reason the world learned of the existence of the Higgs boson only when the hills of data of two independent experiments had both scaled it. The vocabulary of Gauss, born to weigh the mistakes of a Sicilian telescope, has become the liturgy with which humanity decides what exists. The operative vocabulary of this discipline — how uncertainties are composed, how their sources are classified, how the margin of a calibration certificate is written according to the international guides — is matter for the manual, and the professional reader will find it in the entries of the glossary of ingegnerismo.it [ingegnerismo.it] ; here it is the philosophical substance that interests us, which lets itself be said in a line: error has ceased to be a sin of the observer and become a property of the world — measurable, declarable, honourable. The shame that killed Méchain’s peace has overturned into its contrary: today the dishonour is not to have deviations, it is to hide them.

The first chapter had left a question of vocabulary in suspense, and it is the moment to settle it. There it was observed that the premodern world knew the “just” measure and the “false,” but not the “exact,” because the reference against which to err was lacking; now the circle closes in an unexpected way. Mature science has in its turn ceased to call its own measures “exact”: every experimental number carries its margin, and the word survives with a single legitimate technical meaning — a quantity is called exact whose uncertainty is zero by definition, because a convention has fixed it. The inch of twenty-five point four millimetres of the fourth chapter is exact in this sense; and in the next chapter we shall see humanity confer this exactness by decree on a handful of constants of nature. “Exact,” born as the opposite of “wrong,” has ended by meaning “agreed”: the journey of a word that sums up the journey of the book.

And it was with these spectacles that the physics of the twentieth century learned to read even its own constants. In the thirties the American spectroscopist Raymond Birge inaugurated a genre destined for great fortune: to take all the world’s measures of a constant of nature — the speed of light, the charge of the electron — and fuse them with least squares into a “recommended” value, accompanied by its uncertainty and periodically updated. The universal constants, the most objective quantities one can conceive, receive ever since their official value by statistical adjustment of discordant measures: the method born to find a planetoid again governs today the fundamental numbers of the universe, and the reader will find it at work in the next chapter, where one of those adjustments will decide the eternal value of the Planck constant.

With this tool in hand, the story can face its last act. For to declare the uncertainty means also, inevitably, to put the standards on trial: if every measure has its margin, it makes sense to ask how stable the platinum metre is, how much the cylinder shut in the vault of the three keys really weighs, decade after decade. For a century the periodic verifications accumulated numbers in silence; and the numbers, read with the spectacles of this chapter, said something no one wished to hear: the kilogram of the world and its copies were moving apart from one another. By how much, why, and what humanity decided to do about it, is the story of the next chapter — that of a bar of light and a Planck constant that pensioned off the last artefact.